Reduction of elliptic curves over certain real quadratic number fields

The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a a-rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at 2 is allowed. Small fields satisfying the hypotheses are then found, and in four cases the non-existence of such elliptic curves can be shown, while in three others all such curves have been classified.